Assume that \(b(\cdot )\)  is smooth and vanishes in the vicinity of \(\lbrace -1,1\rbrace \) . Additionally, assume \(\Phi (u)\)  is smooth vanishing near the origin and at infinity. For any two different unitary vectors \(\widehat{u},\, \widehat{\xi }\) , define \(\varphi \)  as \(\cos (\varphi )=\widehat{u}\,\cdot \,\widehat{\xi }\) . Here we argue in three dimension for the technical simplicity of exposition, however, these ideas can be extended to \(\mathbb {R}^{d}\) , see [1]}, [2]}. In particular see [3]} for the Boltzmann context. In polar coordinates, we parameterize \(\sigma \in \mathbb {S}^{2}\)  as
\(\sigma = \cos (\theta )\widehat{u} + \sin (\theta )\sin (\phi )\widehat{w}_{1} + \sin (\theta )\cos (\phi )\widehat{w}_{2}\,.\) 
